Vol. 14, No. 3, 2021

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Restriction of toral eigenfunctions to totally geodesic submanifolds

Xiaoqi Huang and Cheng Zhang

Vol. 14 (2021), No. 3, 861–880

We estimate the L2 norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus 𝕋d, d 2. We reduce getting correct bounds to counting lattice points in the intersection of some ν-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain–Rudnick on L2-restriction estimates for rational hyperplanes. On 𝕋2, we prove the uniform L2 restriction bounds for closed geodesics. On 𝕋3 , we obtain explicit L2 restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.

eigenfunction estimates, restriction, geodesic
Mathematical Subject Classification 2010
Primary: 11P21, 35P20, 58J50
Received: 27 February 2019
Revised: 24 September 2019
Accepted: 21 November 2019
Published: 18 May 2021
Xiaoqi Huang
Department of Mathematics
Johns Hopkins University
Baltimore, MD
United States
Cheng Zhang
Department of Mathematics
Johns Hopkins University
Baltimore, MD
United States